# Gauge origin of the Dirac field and singular solutions to the Dirac equation

## TL;DR

This paper explores the gauge origin of the Dirac field, showing how all Dirac solutions relate to Klein-Gordon solutions via differentiation, revealing new solution chains, gauge invariance, and internal symmetries, with extensions to external fields.

## Contribution

It introduces a novel method linking Dirac and Klein-Gordon solutions through gauge transformations, expanding understanding of Dirac solutions and their symmetries.

## Key findings

- All Dirac solutions can be derived from Klein-Gordon solutions.
- Identification of new chains of solutions to Dirac and Klein-Gordon equations.
- Demonstration of gauge invariance and internal symmetries in solutions.

## Abstract

We present and enhance our previous statements (arXiv:0907.4736) on the non-canonical interrelations between the solutions to the free Dirac equation (DE) and the Klein-Gordon equation (KGE). We demonstrate that all the solutions to the DE (singular ones among them) can be obtained via differentiation of a pair of the KGE solutions for a doublet of scalar fields. In this way we obtain a "spinor analogue" of the mesonic Yukawa potential and previously unknown chains of solutions to the DE and KGE. The pair of scalar "potentials" is defined up to a gauge transformation under which the corresponding solution of the DE is invariant. Under transformations of the Lorentz group, canonical spinor transformations form only a subclass of more general transformations of the DE solutions, under these the generating scalar potentials undergo transformations of internal symmetry intermixing their components. In particular, under continuous rotation through one full revolution the transforming solutions, as a rule, take their original values ("spinor two-valuedness" is absent). With an arbitrary solution of the DE one can associate, apart from the standard one, a non-canonical set of conserved quantities, a positive definite "energy" density among others, and with any KGE solution -- a positive definite "probability density", etc. Finally, we discuss a generalization of the proposed procedure to the case when an external electromagnetic field is present.